Can a neural network learn to avoid wrong decisions using backpropagation?
What you are describing is conceptually close to adversarial training. you should read more on adversarial examples and generative adversarial networks for more information.

The idea is that there is a discriminator network, whose job is to correctly discriminate between positive and negative examples. We also have a generative network, that learns to produce "adversarial examples" that "confuses" the discriminator network. By training these two networks side by side, both networks get better at their task. But it's usually the generator network that people are more interested in.

Intuitively, the naive implementation of the method you've described (gradient ascent on incorrect examples from a network in a clean/randomly-initialized state) shouldn't work. This is because negative examples don't form a "natural class" (all triangles have 3 edges, all things that are not triangles however....)